The Extended Block Predictor-Block Corrector Method for Computing
Fuzzy Differential Equations
JIMEVWO GODWIN OGHONYON
Department of Mathematics, Covenant University,
Km 10, Idiroko, Canaan Land,Ota, Ogun State, NIGERIA
MATTHEW ETINOSA EGHAREVBA
Department of Sociology, Covenant University,
Km 10, Idiroko, Canaan Land, Ota, Ogun State, NIGERIA
OGBU FAMOUS IMAGA
Department of Mathematics, Covenant University,
Km 10, Idiroko, Canaan Land, Ota, Ogun State, NIGERIA
Abstract: - Over the years, scholars have developed predictor-corrector method to provide estimates for ordinary
differential equations (ODEs). Predictor-corrector methods have been reduced to predicting-correcting method with
no concern for finding the convergence-criteria for each loop with no suitable vary step size in order to maximize
error. This study aim to consider computing fuzzy differential equations employing the extended block predictor-
block corrector method (EBP-BCM). The method of interpolation and collocation combined with multinomial
power series as the basis function approximation will used. The principal local truncation errors of the block
predictor-block corrector method will be utilized to bring forth the convergence criteria to ensure speedy
convergence of each iteration thereby maximizing error(s). Thus, these findings will reveal the ability of this
technique to speed up the rate of convergence as a result of variegating the step size and to ensure error control.
Some examples will solve to showcase the efficiency and accuracy of this technique.
Keywords: fuzzy differential equations; ebp-bcm; principal local truncation errors; converging-criteria; max-error.
Received: July 5, 2021. Revised: June 7, 2022. Accepted: August 2, 2022. Available online: September 19, 2022.
1 Introduction
Fuzzy differential equations are geared towards
modelling real life problems with incertitude and
fuzziness. For instance, there might be an
impossibility to ascertain precisely the initial value
and such mapping might comprise of unsettled
parametric quantities. This inexactitude extends to
the requirement of fuzzy differential equations
(FDEs) to surmount the problem. This aspect of
fuzzy differential equations takes place in numerous
studies like scientific discipline, economic science,
psychological science, defense mechanism, human
ecology and applied sciences [17], [23], [36] and
[39]. Scholars like [9] were the foremost to present
fuzzy differential equations. The general idea was
unfolded by [11], while authors like [7], [8], [16] and
[33] analyze and suggest various essential
explanations including the propositions in the fuzzy
differential. Others importantly contributed in the
areas of H-derivative, Hukuhara differentiable, fuzzy
initial value problems (FIVPs) and its generality. See
[8], [18], [19] and [34].
Over the years, several research studies on
fuzzy differential equations have been executed by
scholars adopting numerical methods of single-step
methods, multistep methods, predictor-corrector
system, Leffler kernel differential operator of the
ABC approach, reproducing kernel method (RKM),
reproducing kernel Hilbert space method (RKHSM),
reproducing kernel theory (RKT) The ideas behind
numerical technics are due to the exact results of
some FDEs been disadvantage and windy to find out
on account of the ramification of the FDEs. Various
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bookmen talked about and employ the Adams family
of predictor-corrector pair for resolving FDEs [4],
[5], [7], [12], [13], [14], [17], [20], [36] and [39].
Although, all of these methods resolve to predict and
correct the result without a desirable step size to
decide the convergence to maximize error with error
control. Again, the natures of fuzzy differential
equations are exponential in nature with stiff
properties. As a result of the incertitude and fuzziness
of the differential equations, the numerical technics
mentioned earlier lacks the capacity to yield a better
result and stabilize the looping via providing a
suitable step size to decide the convergence-criteria
thereby yielding a better max error. Nevertheless,
EBP-BCM has the capacity to vary the step-with
same order and utilize a suitable steps size. These
ideas used will stabilize the convergence criteria and
foster a faster rate of looping with a better max-error.
EBP-BCM has an advantage over other numerical
methods because of its ability to solve stiff problem
using the approach of vary step of the same order
and finding a suitable step size. See [1], [2],[22],[23],
[24], [27], [28], [29], [30] and [31].
This study is motivated by need to build a
block numerical model with high performance ability
to solve problem whose analytical solution is
exponential in nature with stiffness properties. Thus,
EBP-BCM possesses the ability to utilize variable
step with same order and suitable step size to resolve
exponential solution. This concept will satisfy the
convergence criteria and yield lesser max-errors.
The contribution and novelty of this research
study will be to introduce the concept of variable
step, same order and suitable variable step size as a
primal tool to bring about speedy convergence, error
control and lesser max-error(s). The convergence of
each tolerance is dependent on the ability to
determine a suitable step size for each loop process.
The contribution and novelty of the
extension of block predictor-block corrector method
is the ability to find a suitable step size to satisfy each
convergence criteria for every loop of the iteration
thereby leading to better efficiency and accuracy.
Again, these ideas have been extended to computing
the fuzzy differential equations to yield better results
compare to others.
This study is coordinated as follows:
Introductory explanation and notational systems will
be discussed in part 1. Materials and methods of the
block extended predictor-block corrector method will
be described by fuzzy in part 2. Results and
discussion will be established in part 3 and finally,
the conclusion will be summarized at the termination
of the study in part 4. See [3] and [17].
1.1 Preliminaries
The introductory explanations and notational systems
that is very important for this study will be critically
appraised in this part. For more action, see [17], [35],
[36] and [39].
Definition 1: Consider is a
single-valued function number with R as a setup of
the entire real numbers. The single-valued function
possesses the next attributes:
i. is defined as the upper semi continuous,
ii. is set as the fuzzy convex, that is
,
iii. is the normal, i. e. there exist
iv. The stand of is sup
and its law of closure lcl (sup) is compact.
Assume E is a setup of the entire fuzzy
numbers on R See [17], [35], [36] and [39].
Definition 2: Consider a fuzzy number in
parametric pattern is seen as a pair of of
mappings , which meets the
next necessities:
i. is a bounded monotonic non-decreasing
left uninterrupted mapping.
ii. is a bounded monotonic non increasing
left uninterrupted mapping.
iii. .
Assume is a real time interval. A single-valued
function is denoted as a fuzzy physical
process and its set is called by
.
(1)
See [17], [35], [36] and [39].
Definition 3: Consider
constitute a fuzzy set on , since it
is referred to triangular fuzzy number assume its
membership mapping constituted
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(2)
See [17], [32], [33] and [35].
Definition 4: The triangular fuzzy number is
a given fuzzy set that portray the character of
an ordered three fold with
and , where
set of a triangular fuzzy number is
established by
, for whenver . (3)
See [17], [35], [36] and [39].
Remark 1: Consider is seen to
possess Hukuhara derivative and its Hukuhara
differential coefficient is integrable , in that
case
(4)
Definition 5: A single-valued function
is referred to as fuzzy physical process. This is
indicated as
.
(5)
The Seikkala differential coefficient of a fuzzy
physical process is set by
,
(6)
with the understanding that this par in reality
determines a fuzzy number .
Remark 2: Consider: is seen as
Seikkala derivative and its Seikkala differential
coefficient is integrable throughout , so
(7)
See [17], [34], [35], [36] and [39].
1.2 Fuzzy Differential Equations
For this study, we examine the initial value problem
where is a continuous function of into
and , represent fuzzy numbers defined
in set E. From above, the order fuzzy differential
equation will be determined by varying quantities
,
which translates to the complying fuzzy system
,
,
,
where define the continuous function of
into and
represent fuzzy numbers in
set E with level intervals.
Again, we examine the next fuzzy initial
value problem (FIVP) , where is a
fuzzy mapping of , is a fuzzy mapping of the
crisp variable and the fuzzy physical variable and
is called the Hukuhara or Seikkala fuzzy
differential coefficient of . Consider the fuzzy
Cauchy problem
, ,
(8)
See [35].
The state of the existing theorem is
discovered for the Cauchy physical problem (8). See
[17], [35], [36] and [39].
Consider assume
is seen as the Hukuhara derivative, so
. Then (8) interprets
into the next physical system of ordinary differential
equations (ODEs):
,
,
(9)
,
See [17], [35], [36] and [39].
Theorem 1: Consider
constitute to ascertain the data and presuppose that
for each one of the . So for
each one be
an element in . This shows that
,
,
(10)
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where
See [17], [35], [36], and [39].
2 Formulating the Block Predictor-
Block Corrector Pair
This section considers formulating the block
predictor-block corrector pair of the fuzzy
differential equations. The choice of the block
predictor-block corrector originates from [23] which
suggest that the key to greater efficiency and
accuracy is the implementation of variable step,
variable order and variable step size. This study
intends to implement variable step, same order and
variable step size. This will be clearly defined in the
block predictor-block corrector method.
Note: This aspect characterizes Cauchy fuzzy
differential equations
,
.
Suppose has differentiability then in
that case . Thus, it
follows right away
,
,
.
Again, suppose possess
differentiability and then
. Thusly, it agrees at once
,
.
Writing the block predictor method in the form of
(10) as shown below. Setting as a point of
interpolation and as the given point of
collocations. The fuzzy initial values are constituted
as , . The block predictor method is
constructed utilizing as interpolation, while
represent points of
collocation. The evaluated values are ,
, . Likewise, the block
corrector method is developed using
,
to constitute collocation points. The estimated values
are , , . This
unification will form
fy im
k
iiim
iihty
1
1
1
1
1
)(
,
f
h
yim
k
iiim
ii
ty
1
0
1
1
1
0
)(
.
(11)
Exploring (10) to interpolation and collocation
results to ,
for block predictor method and
,
for block corrector method. See [17],
[26], [27], [28], [29], [30], [31], [34], [35], [36] and
[39].
From (10), the fuzzy combination of (11) gives
+
+
.
Given the basis function approximation
,
for of the block predictor method, the
point of interpolation and collocations are derived as:
,
,
,
.
Similarly, for of block corrector
method, the point of interpolation and collocations
are generated as:
,
,
,
.
Consequently, conforming to the above interpolated
and collocated results will bring forth the block
predictor- block corrector method as
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,
,
.
,
,
.
Combining the points of interpolation and collocation
derived above will give rise to form . Noting
that the block predictor method is ,
, while the block corrector
method is , ,
. See [17], [26], [27], [28], [29], [30], [31], [34],
[35], [36] and [39] for more details.
Recalling fuzzy process and Seikkala
derivative of the fuzzy process is outlined as
, ,
, .
From the above, these results are true for block
predictor method:
, ,
, ,
,
,
,
,
and block corrector method:
, ,
,
,
,
,
,
.
Applying the Hukuhara differentiability of
then ,
the following results are acceptable as
,
,
,
,
,
,
,
and block corrector method:
,
,
,
,
,
,
. See
[17], [24], [25], [26], [27], [28], [29], [31], [32], [33]
and [35] for more particulars.
Computing the generated results of
Seikkala/Hukuhara differentiability and substituting
into the basis function will result to the extended
block predictor-block method. By adopting fuzzy
Hukuhara differentiability process, the block
predictor method for computing fuzzy differential
equations is formed as:
,
,
,
,
(12)
,
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In the same manner, the block corrector method is
generated as
,
,
,
(13)
,
,
.
Equations (12) and (13) are called the extended block
predictor- block corrector method of the fuzzy
differential equations. See [17], [26], [27], [28], [29],
[30], [31], [34], [35], [36] and [39] for more
reference.
2.1 Implementing the Convergence-Criteria
of the Extended Block Predictor-Block
Corrector Method
To drive this process to a logical conclusion, the w-
step block predictor method and w-1-step block
corrector method must have similar order. See [6],
[10], [15], [22], [23], [26], [27], [28], [29], [30] and
[31] for more. Linking [6], [10], [15], [22], [23],
[26], [27], [28], [29], [30] and [31]. It is executable to
determine an approximative principal-local-
truncation-error of the extended block predictor-
block corrector method voiding loftier derivatives,
. Nevertheless, for , where and
defines the order of the extended block predictor-
block corrector method. Quickly, for method of order
defines the order and analyzing the block
predictor method of to generate the
principal-local-truncation-errors as
(14)
Equation (14) complies with the fuzzy process of
Hukuhara differentiability derivatives.
In similar manner, examining the block corrector
method of brings about the principal-
local-truncation-error as
(15)
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where
and
proceeds as the sorted physical
quantities of and work as
the analytical results of the loftier derivatives which
function in conformity to the initial assumptions
, . See [6], [10], [15], [22],
[23], [24], [25], [26], [27], [28], [29], [31], [32], [33],
[34] and [35].
In moving forward, for small presumption values of
is ascertained as
, ,
where the strength of the extended block predictor-
block corrector counts on this presumptuousness
submitted before. Further step-down of the principal-
local-truncation-errors of (14) and (15) above
likewise voiding preconditions of order ,
. Therefore, this gives no challenge
reaching the computational output of the principal-
local-truncation errors of the extended block
predictor-block corrector method
(16)
.
Concern asserting the existence of the truth that
,
,
,
,
,
defines the
predicting and correcting approximations instituted
via the extended block predictor-block corrector
method of order , despite
,
,
,
,
and
are distinctively referred
to as principal-local-truncation-errors. ,, ,,
, are the convergence-criteria of the extended
block predictor-block corrector method.
Setting ahead, these approximative of the principal-
local-truncation-errors (16) are used to make vital
conclusion on the iteration results to either accept or
remodel the looping with a smaller step size. The
acceptability of the step size is established on an
experiment placed by (16). See [6], [10], [15], [22],
[23], [26], [27], [28], [29], [30], [31], [34], [35], [36],
[38] and [39] for more items.
The principal-local-truncation-errors of the
converging-criteria is distinctively the extended
block predictor-corrector method aligning to
convergence which acts as the stabilizer of the
computed results.
3 Numerical Problems of Fuzzy
Differential Equations
Two fuzzy differential equations were solved using
EBP-BCM. The two problems of fuzzy differential
equations are exponential in nature with stiff
properties. The stiffness properties of the fuzzy
differential equations engenders the use of vary step
with same order and suitable variable step size
approach. The convergence-criteria of , ,
, and were utilized. See [16] and
[31]. A written computer-code in Mathematica format
is carried-out on the platform of Mathematica 9
kernel. See [1], [2], [17] and [36].
Example 1
Study the linear fuzzy differential equations:
,
.
Exact solutions
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,
,
.
Example 2
Consider un-dimensional fuzzy differential
equations:
.
Exact solutions
,
,
.
4 Results and Discussion
Table I and Table II displays the computed results
carried out on two fuzzy differential equations. The
implementation is executed on Mathematica 9 Kernel
to quicken and easy computation burden in order to
demonstrate the potency and proficiency of EBP-
BCM. The implementation of the EBP-BCM starts
by combining the block predictor-block corrector
method together with the converging criteria to find a
worthy step size that will bring about speedy
convergence and maximize error. Again, to achieve
faster convergence with better efficiency and
accuracy, a suitable step is found for each EBP-BCM
to satisfy the convergence criteria. This process is
carried out repeatedly until the convergence criteria
are met. Fuzzy differential equations have analytical
solutions that are exponential in nature and as such
EBP-BCM is required for better results. See [1], [2],
[17] and [36]. Table 1 and Table 2 presents the
summary as follows
Table 1. Numerical Results for Example 1
Mutized Maxerrs()
PC 0.001112770464
PC
PC
EBP-BCM
EBP-BCM
EBP-BCM
Mutzed Maxerrs()
PC 0.001112770464
PC
PC
EBP-BCM
EBP-BCM
EBP-BCM
Mutzed Maxerrs()
PC 0.000898657928
PC
PC
EBP-BCM
EBP-BCM
EBP-BCM
Mutized Maxerrs()
PC
PC
PC
EBP-BCM
EBP-BCM
EBP-BCM
Mutzed Maxerrs()
PC
PC
PC
EBP-BCM
EBP-BCM
EBP-BCM
Mutzed Maxerrs()
IPC
IPC
IPC
EBP-BCM
EBP-BCM
EBP-BCM
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Table 2. Numerical Results for Example 2
Mutzed Maxerrs()
2PDO4
2PDO4
2PDO4
EBP-BCM
EBP-BCM
EBP-BCM
ERK41
ERK41
ERK41
RK41
RK41
RK41
EBP-BCM
EBP-BCM
EBP-BCM
ANN1
ANN1
ANN1
EBP-BCM
EBP-BCM
EBP-BCM
Mutzed Maxerrs()
2PDO4
2PDO4
2PDO4
EBP-BCM
EBP-BCM
EBP-BCM
ERK41
ERK41
ERK41
RK41
RK41
RK41
EBP-BCM
EBP-BCM
EBP-BCM
ANN1
ANN1
ANN1
EBP-BCM
EBP-BCM
EBP-BCM
4.1 Nomenclature
The word form stated on Table I and Table 2 are
defined as follows:
EBP-BCM: calculated-max-errors in EBP- BCM
(extended
block predictor-block corrector method) for example
1 and 2.
: represents fuzzy counts having restricted
Mutzed : method put to use.
Maxerrs(): mag. of lower calculated-max-errors
in EBP-BCM.
Maxerrs(): mag. of average calculated-max-
errors in EBP-BCM.
Maxerrs(): mag. of upper calculated-max-errors
in EBP-BCM.
: converging-criteria.
PC ((): mag. of lower calculated max-errors in
PC
(predictor-corrector algorithm of ,) for
example 1.
See [36].
PC (): mag. of average calculated max-errors
in PC
(predictor-corrector algorithm of ,) for
example 1.
See [36].
PC (): mag. of lower calculated max-errors in
PC
(predictor-corrector algorithm of ,) for
example 1
See [36].
IPC ((): mag. of lower calculated max-errors
in PC
(predictor-corrector algorithm of ,) for
example 1.
See [36].
IPC (): mag. of average calculated max-errors
in PC
(predictor-corrector algorithm of ,) for
example 1.
See [36].
IPC (): mag. of lower calculated max-errors in
PC
(predictor-corrector algorithm of ,) for
example 1.
See [36].
ANN((): mag. of lower calculated max-errors
in
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ERK4 (artificial neural network approach of )
for
example 2. See [17].
ANN (): mag. of upper calculated max-errors
in ERK4
(artificial neural network approach of )for
example 2.
See [17].
ERK4((): mag. of lower calculated max-errors
in
ERK4 (extended Runge-Kutta formulae of order 4 of
)
for example 2. See [17].
ERK4 (): mag. of upper calculated max-errors
in
ERK4 (extended Runge-Kutta formulae oforder 4 of
)
for example 2. See [17].
RK4((): mag. of lower calculated max-errors
in RK4
(Runge-Kutta formulae of order 4 of )
forexample 2.
See [17].
RK4 (): mag. of upper calculated max-errors
in RK4
(Runge-Kutta formulae of order 4of ) for
example 2.
See [17].
2PDO4((): mag. of lower calculated max-
errors in
2PDO4 (2-point diagonally implicit multistep method
of
order four of ) for example 2. See [17].
2PDO4 (): mag. of upper calculated max-
errors in
2PDO4 (2-point diagonally implicit multistep method
of
order four of ) for example 2. See [17].
5 Conclusion
The result presented that EBP-BCM is achievable
applying the convergence-criteria. This convergence-
criterion resolves the toleration or non-toleration of
the loop with suited/variegating step size. The final
result shows the functioning and accuracy in terms of
the magnitude define as (), () and
() (lower, average and upper max errors at all
levels of , , , and . The
possibility of this research study is done via working-
out a worthier step size that will satisfy the
convergence-criteria which will bring about the
needed results. Thusly, the EBP-BCM is seen to do
better than PC, IPC ANN, ERK4, RK4 and 2PDO
compare to others without these vantages stated
above. Lastly, this procedure of utilizing the EBP-
BCM generates a faster method of looping and
stability towards a better accuracy. The Mathematica
Kernel is utilized to preserve the time of
computation, space and memory.
Future Work:
The future work will be to develop a block predictor-
block corrector method of variable step-variable
order-suitable variable step size to compute fuzzy
differential equations with exponential and
trigonometry solutions.
Acknowledgements:
Bookmen sincerely appreciate Covenant University
for providing sponsorship throughout this project.
My thanks are due to all anonymous reviewers for
their guidance and immense contributions towards
this project.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.1
Jimevwo Godwin Oghonyon,
Matthew Etinosa Egharevba, Ogbu Famous Imaga
E-ISSN: 2224-2880
11
Volume 22, 2023
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Conflicts of Interest:
The authors reiterate that there are no conflicts of
interest regarding the publication of this article and
among the authors.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Jimevwo Godwin Oghonyon propounded the
concept, methodology and executed the problem
using Mathematica.
-Matthew Etinosa Egharevba carried out the proof
reading and supervision.
-Imaga Ogbu Famous ensure continuous support via
logistics and technical support.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research project is funded by Covenant
University, Ota,Nigeria.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_
US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.1
Jimevwo Godwin Oghonyon,
Matthew Etinosa Egharevba, Ogbu Famous Imaga
E-ISSN: 2224-2880
12
Volume 22, 2023
